For example an inequality going back to Pólya (5.4 in [1]} for the planar case) asserts that for any open set with finite measure the product of torsional rigidity and principal Dirichlet eigenvalue is bounded by measure. In [2]} it was shown that this inequality is sharp but that no optimal set exists. A more general functional was studied in [3]}. There the principal eigenvalue with the \(q\) -'th power of the torsional rigidity was studied among open sets with fixed Lebesgue measure. For \(0<q<2/(d+2)\)  the principal eigenvalue dominates the behaviour, and the ball is the minimiser.
On the other hand for \(q>1\)  the torsional rigidity dominates the behaviour, and the functional has a maximiser in the class of open, bounded convex sets. Recently the functionals involving powers of the perimeter and torsional rigidity have been studied under a fixed measure constraint, [4]} and [5]}. In order to guarantee well-posedness of the problem it suffices to consider the collection of non-empty, open, bounded and convex sets. If \(d=2\)  then that convexity constraint could be relaxed by considering non-empty, open, bounded Lipschitz sets with a topological constraint instead, [5]}.
